Handout 3. Free Electron Gas in 2D and 1D
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- Godwin Montgomery
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1 Handout 3 F lcton Gas in D and D In this lctu ou will lan: F lcton gas in two dinsions and in on dinsion Dnsit o Stats in -spac and in ng in low dinsions C 47 Sping 9 Fahan Rana Conll Univsit lcton Gass in D In sval phsical ssts lcton a conind to ov in just dinsions apls discussd in dtail lat in th cous a shown blow: STM icogaph Siconducto Quantu Wlls: Gaphn: Gas InGas quantu wll (- n) Gas Siconducto quantu wlls can b coposd o ptt uch an siconducto o th goups II III IV V and VI o th piodic tabl TM icogaph Gaphn is a singl atoic la o cabon atos aangd in a honcob lattic C 47 Sping 9 Fahan Rana Conll Univsit
2 lcton Gass in D In sval phsical ssts lcton a conind to ov in just dinsion apls discussd in dtail lat in th cous a shown blow: Siconducto Quantu Wis (o Nanowis): Siconducto Quantu Point Contacts (lctostatic Gating): Cabon Nanotubs (Rolld Gaphn Shts): Gas tal tal InGas Nanowi InGas Quantu wll Gas Gas C 47 Sping 9 Fahan Rana Conll Univsit lctons in D Mtals: Th F lcton Modl Th quantu stat o an lcton is dscibd b th ti-indpndnt Schoding quation: V Consid a lag tal sht o aa = : Us th Sold odl: Th lctons insid th sht a conind in a two-dinsional ininit potntial wll with zo potntial insid th sht and ininit potntial outsid th sht V o insid th sht V o outsid th sht Th lcton stats insid th sht a givn b th Schoding quation lctons (pinc no potntial whn insid th sht) C 47 Sping 9 Fahan Rana Conll Univsit
3 3 C 47 Sping 9 Fahan Rana Conll Univsit Bon Von Kaan Piodic Bounda Conditions in D Solv: Us piodic bounda conditions: z z z z Ths ipl that ach dg o th sht is oldd and joind to th opposit dg Solution is: i i. Th bounda conditions dictat that th allowd valus o and a such that: i i n n = ± ± ±3. = ± ± ±3. C 47 Sping 9 Fahan Rana Conll Univsit Bon Von Kaan Piodic Bounda Conditions in D abling Sch: ll lcton stats and ngis can b labld b th cosponding -vcto i. Montu ignstats: noth advantag o using th plan-wav ng ignstats (as opposd to th sin ng ignstats) is that th plan-wav stats a also ontu ignstats Montu opato: i p ˆ i p ˆ Noalization: Th wavunction is popl noalizd: d Othogonalit: Wavunctions o two dint stats a othogonal: i d d '. ' * ' Vlocit: Vlocit o ignstats is: v
4 Stats in D -Spac -spac Visualization: Th allowd quantu stats stats can b visualizd as a D gid o points in th nti -spac n n = ± ± ±3. Dnsit o Gid Points in -spac: ooing at th igu in -spac th is onl on gid point in v sall aa o siz: Th a gid points p unit aa o -spac V ipotant sult C 47 Sping 9 Fahan Rana Conll Univsit Th lcton Gas in D at Zo Tpatu - I Suppos w hav N lctons in th sht. Thn how do w stat illing th allowd quantu stats? Suppos T~K and w a intstd in a illing sch that givs th lowst total ng. N Th ng o a quantu stat is: Statg: ach gid-point can b occupid b two lctons (spin up and spin down) Stat illing up th gid-points (with two lctons ach) in cicula gions o incasing adii until ou hav a total o N lctons Whn w a don all illd (i.. occupid) quantu stats cospond to gid-points that a insid a cicula gion o adius F F C 47 Sping 9 Fahan Rana Conll Univsit 4
5 Th lcton Gas in D at Zo Tpatu - II ach gid-point can b occupid b two lctons (spin up and spin down) ll illd quantu stats cospond to gid-points that a insid a cicula gion o adius F a o th cicula gion = F Nub o gid-points in th cicula gion = F F Fi cicl Nub o quantu stats (including spin) in th cicula gion = F F But th abov ust qual th total nub N o lctons insid th bo: N F n lcton dnsit F n N F Units o th lcton dnsit n a #/c C 47 Sping 9 Fahan Rana Conll Univsit Th lcton Gas in D at Zo Tpatu - III ll quantu stats insid th Fi cicl a illd (i.. occupid b lctons) ll quantu stats outsid th Fi cicl a pt Fi Montu: Th lagst ontu o th lctons is: F This is calld th Fi ontu Fi ontu can b ound i on nows th lcton dnsit: n F Fi ng: Th lagst ng o th lctons is: This is calld th Fi ng F : F F F F Fi cicl lso: F n o n F Fi Vlocit: Th lagst vlocit o th lctons is calld th Fi vlocit v F : v F F C 47 Sping 9 Fahan Rana Conll Univsit 5
6 Th lcton Gas in D at Non-Zo Tpatu - I Rcall that th a gid points p unit aa o - spac d d So in aa d d o -spac th nub o gid points is: d d d Th suation ov all gid points in -spac can b placd b an aa intgal d all Tho: d N all is th occupation pobabilit o a quantu stat C 47 Sping 9 Fahan Rana Conll Univsit Th lcton Gas in D at Non-Zo Tpatu - II Th pobabilit that th quantu stat o wavvcto is occupid b an lcton is givn b th Fi-Diac distibution unction: KT Wh: Tho: N d d KT Dnsit o Stats: Th -spac intgal is cubso. W nd to convt into a sipl o an ng spac intgal using th ollowing stps: d d and d d Tho: d d d C 47 Sping 9 Fahan Rana Conll Univsit 6
7 Th lcton Gas in D at Non-Zo Tpatu - III d N d g D KT KT g D Dnsit o stats unction is constant (indpndnt o ng) in D g D () has units: # / Joul-c Wh: Th poduct g() d psnts th nub o quantu stats availabl in th ng intval btwn and (+d) p c o th tal Suppos cosponds to th inn cicl o th lation: nd suppos (+d) cosponds to th out cicl thn g D () d cosponds to twic th nub o th gid points btwn th two cicls C 47 Sping 9 Fahan Rana Conll Univsit Th lcton Gas in D at Non-Zo Tpatu - IV N d gd d g D KT g D Wh: g D Th pssion o N can b visualizd as th intgation ov th poduct o th two unctions: Chc: Suppos T=K: N d gd T = K n Copa with th pvious sult at T=K: n F d g D t T=K (and onl at T=K) th Fi lvl is th sa as th Fi ng F C 47 Sping 9 Fahan Rana Conll Univsit 7
8 Fo T K: Th lcton Gas in D at Non-Zo Tpatu - V Sinc th cai dnsit is nown and dos not chang with tpatu th Fi lvl at tpatu T is ound o th pssion n d gd K T K T log KT In gnal th Fi lvl is a unction o tpatu and dcass o F as th tpatu incass. Th act lationship can b ound b invting th abov quation and calling that: to gt: T n F F KT log KT C 47 Sping 9 Fahan Rana Conll Univsit Total ng o th D lcton Gas Th total ng U o th lcton gas can b wittn as: d U all Convt th -spac intgal to ng intgal: U d g U D Th ng dnsit u is: u d g D Suppos T=K: F u d gd F Sinc: n F W hav: u n F C 47 Sping 9 Fahan Rana Conll Univsit 8
9 9 C 47 Sping 9 Fahan Rana Conll Univsit D lcton Gas in an pplid lctic Fild - I lcton distibution in -spac whn -ild is zo lcton distibution is shitd in -spac whn -ild is not zo ˆ t Distibution unction: Distibution unction: Sinc th wavvcto o ach lcton is shitd b th sa aount in th psnc o th -ild th nt ct in -spac is that th nti lcton distibution is shitd as shown C 47 Sping 9 Fahan Rana Conll Univsit D lcton Gas in an pplid lctic Fild - II lcton distibution is shitd in -spac whn -ild is not zo Distibution unction: Cunt dnsit (units: /c) v d J Do a shit in th intgation vaiabl: n J d J d J v d J Wh: n Sa as th Dud sult - but units a dint. Units o a Sins in D lcton dnsit = n (units: #/c )
10 lctons in D Mtals: Th F lcton Modl Th quantu stat o an lcton is dscibd b th ti-indpndnt Schoding quation: V Consid a lag tal wi o lngth : Us th Sold odl: Th lctons insid th wi a conind in a on-dinsional ininit potntial wll with zo potntial insid th wi and ininit potntial outsid th wi V V o insid th wi o outsid th wi Th lcton stats insid th wi a givn b th Schoding quation lctons (pinc no potntial whn insid th wi) C 47 Sping 9 Fahan Rana Conll Univsit Solv: Bon Von Kaan Piodic Bounda Conditions in D Us piodic bounda conditions: z z Solution is: i Ths ipl that ach act o th sht is oldd and joind to th opposit act Th bounda conditions dictat that th allowd valus o a such that: n i n = ± ± ±3. C 47 Sping 9 Fahan Rana Conll Univsit
11 Stats in D -Spac -spac Visualization: Th allowd quantu stats stats can b visualizd as a D gid o points in th nti -spac n n = ± ± ±3. Dnsit o Gid Points in -spac: ooing at th igu in -spac th is onl on gid point in v sall lngth o siz: Th a gid points p unit lngth o -spac V ipotant sult C 47 Sping 9 Fahan Rana Conll Univsit Th lcton Gas in D at Zo Tpatu - I ach gid-point can b occupid b two lctons (spin up and spin down) ll illd quantu stats cospond to gid-points that a within a distanc F o th oigin ngth o th gion = F F F Nub o gid-points in th gion = F Fi points Nub o quantu stats (including spin) in th gion = F But th abov ust qual th total nub N o lctons in th wi: N F n lcton dnsit n F N F C 47 Sping 9 Fahan Rana Conll Univsit Units o th lcton dnsit n a #/c
12 Th lcton Gas in D at Zo Tpatu - II ll quantu stats btwn th Fi points a illd (i.. occupid b lctons) ll quantu stats outsid th Fi points a pt Fi Montu: Th lagst ontu o th lctons is: F This is calld th Fi ontu Fi ontu can b ound i on nows th lcton dnsit: n F Fi ng: Th lagst ng o th lctons is: F This is calld th Fi ng F : F F lso: n F 8 8 o n F Fi points Fi Vlocit: Th lagst vlocit o th lctons is calld th Fi vlocit v F : v F F C 47 Sping 9 Fahan Rana Conll Univsit Th lcton Gas in D at Non-Zo Tpatu - I Rcall that th a gid points p unit lngth o - spac d So in lngth d gid points is: o -spac th nub o d Th suation ov all gid points in -spac can b placd b an intgal Tho: N all all d d is th occupation pobabilit o a quantu stat C 47 Sping 9 Fahan Rana Conll Univsit
13 Th lcton Gas in D at Non-Zo Tpatu - II Th pobabilit that th quantu stat o wavvcto is occupid b an lcton is givn b th Fi-Diac distibution unction: K T Tho: d N Wh: d KT Dnsit o Stats: Th -spac intgal is cubso. W nd to convt into a sipl o an ng spac intgal using th ollowing stps: d d and d d Tho: d d C 47 Sping 9 Fahan Rana Conll Univsit Th lcton Gas in D at Non-Zo Tpatu - III d N d g D KT KT Wh: g D g D () has units: # / Joul-c Th poduct g() d psnts th nub o quantu stats availabl in th ng intval btwn and (+d) p c o th tal Dnsit o stats unction in D Suppos cosponds to th inn points o th lation: nd suppos (+d) cosponds to th out points thn g D () d cosponds to twic th nub o th gid points btwn th points (adding contibutions o both sids) C 47 Sping 9 Fahan Rana Conll Univsit 3
14 Wh: Th lcton Gas in D at Non-Zo Tpatu - IV N d gd Th pssion o N can b visualizd as th intgation ov th poduct o th two unctions: Chc: Suppos T=K: d g g D KT T = K N d gd Copa with th pvious sult at T=K: n 8 F D 8 n 8 g D d g D t T=K (and onl at T=K) th Fi lvl is th sa as th Fi ng F C 47 Sping 9 Fahan Rana Conll Univsit Fo T K: Th lcton Gas in D at Non-Zo Tpatu - V Sinc th cai dnsit is nown and dos not chang with tpatu th Fi lvl at tpatu T is ound o th pssion n d gd KT In gnal th Fi lvl is a unction o tpatu and dcass o F as th tpatu incass. C 47 Sping 9 Fahan Rana Conll Univsit 4
15 Total ng o th D lcton Gas Th total ng U o th lcton gas can b wittn as: U all d Convt th -spac intgal to ng intgal: U d g u U D Th ng dnsit u is: d g Suppos T=K: D F u d gd 3 8 F 3 Sinc: n 8 F W hav: u n F 3 C 47 Sping 9 Fahan Rana Conll Univsit D lcton Gas in an pplid lctic Fild - I t ˆ lcton distibution in -spac whn -ild is zo Distibution unction: lcton distibution is shitd in -spac whn -ild is not zo Distibution unction: Sinc th wavvcto o ach lcton is shitd b th sa aount in th psnc o th -ild th nt ct in -spac is that th nti lcton distibution is shitd as shown C 47 Sping 9 Fahan Rana Conll Univsit 5
16 6 C 47 Sping 9 Fahan Rana Conll Univsit D lcton Gas in an pplid lctic Fild - II lcton distibution is shitd in -spac whn -ild is not zo Distibution unction: Cunt (units: ) v d I Do a shit in th intgation vaiabl: n I d I d I v d I Wh: n Sa as th Dud sult - but units a dint. Units o a Sins-c in D lcton dnsit = n (units: #/c) ˆ C 47 Sping 9 Fahan Rana Conll Univsit
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